1
Question
If A=⎡⎢⎣10−2−2−22341⎤⎥⎦
such that A−1=pA2+qA+rI, then(p,q,r)=
such that A−1=pA2+qA+rI, then(p,q,r)=
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Solution
The correct option is D (−16,0,56)
A2=⎡⎢⎣−5−8−48122−2−43⎤⎥⎦A3=⎡⎢⎣−10−10−10−16101520−1⎤⎥⎦
A−1=pA2+qA+rI⇒I=pA3+qA2+rA
⇒⎡⎢⎣100010001⎤⎥⎦=p⎡⎢⎣−10−10−10−16101520−1⎤⎥⎦+q⎡⎢⎣−5−8−48122−2−43⎤⎥⎦ +r⎡⎢⎣10−2−2−22341⎤⎥⎦
1=−p−5q+r ...(1)0=0×p−8q+0×r ...(2)0=−10p−4q−2r ...(3)
Solving eqn (1),(2) and (3), we get
p=−16, q=0, r=56
A2=⎡⎢⎣−5−8−48122−2−43⎤⎥⎦A3=⎡⎢⎣−10−10−10−16101520−1⎤⎥⎦
A−1=pA2+qA+rI⇒I=pA3+qA2+rA
⇒⎡⎢⎣100010001⎤⎥⎦=p⎡⎢⎣−10−10−10−16101520−1⎤⎥⎦+q⎡⎢⎣−5−8−48122−2−43⎤⎥⎦ +r⎡⎢⎣10−2−2−22341⎤⎥⎦
1=−p−5q+r ...(1)0=0×p−8q+0×r ...(2)0=−10p−4q−2r ...(3)
Solving eqn (1),(2) and (3), we get
p=−16, q=0, r=56
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